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Mirrors > Home > ILE Home > Th. List > pnfnre | Unicode version |
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
pnfnre |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 7965 |
. . . . . 6
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2 | 1 | uniex 4455 |
. . . . 5
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3 | pwuninel2 6307 |
. . . . 5
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4 | 2, 3 | ax-mp 5 |
. . . 4
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5 | df-pnf 8024 |
. . . . 5
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6 | 5 | eleq1i 2255 |
. . . 4
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7 | 4, 6 | mtbir 672 |
. . 3
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8 | recn 7974 |
. . 3
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9 | 7, 8 | mto 663 |
. 2
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10 | 9 | nelir 2458 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-un 4451 ax-cnex 7932 ax-resscn 7933 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-nel 2456 df-rex 2474 df-rab 2477 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-uni 3825 df-pnf 8024 |
This theorem is referenced by: renepnf 8035 nn0nepnf 9277 xrltnr 9809 pnfnlt 9817 xnn0lenn0nn0 9895 inftonninf 10472 pcgcd1 12360 pc2dvds 12362 |
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